NumPy · Function

np.linalg.qr

Computes the QR factorization of an array, expressing it as an orthonormal factor Q and an upper-triangular factor R. The mode selects reduced, complete, R-only, or raw Householder output.

How it works

The function factors an array into an orthonormal component Q and an upper-triangular component R. The mode controls whether it returns reduced or complete factors, only R, or the raw Householder representation; arrays with leading dimensions are treated as stacks of matrices.

Syntax

numpy.linalg.qr(a, mode='reduced')

Parameters

Name	Type	Description
`a`	`array_like`	An array-like object with at least two dimensions and shape (..., M, N).
`mode`	`{'reduced', 'complete', 'r', 'raw'}, optional`	One of 'reduced', 'complete', 'r', or 'raw'; defaults to 'reduced'.

Returns

For reduced or complete modes, returns Q and R. For r mode, returns R only. For raw mode, returns (h, tau), where h contains Householder reflectors and tau contains their scaling factors.

Key points

Use numpy.linalg.qr to factor a matrix into orthonormal and upper-triangular components.
The defining relation is A = Q @ R, with Q orthonormal and R upper triangular.
Omitting mode selects reduced factorization.
Complete mode expands Q to M columns and returns an M-by-N triangular factor.
The r mode returns only the upper-triangular factor, using K = min(M, N).
Raw output includes tau values that scale the Householder reflectors.

Common mistakes

With mode='raw', the first returned array is a reflector representation rather than Q itself; tau supplies its scaling factors.
The input must have at least two dimensions; a one-dimensional vector is not a valid matrix input.

Example: Reduced QR factorization

import numpy as np

A = np.array([[1., 2.], [3., 4.], [5., 6.]])
Q, R = np.linalg.qr(A)

assert np.allclose(A, Q @ R)
assert np.allclose(Q.T @ Q, np.eye(2))
print(Q.shape, R.shape)

Computes a reduced QR factorization and verifies reconstruction and orthonormal columns.

When to use it

Least-squares workflows

Solve or analyze least-squares systems using an orthonormal-triangular decomposition.

Basis construction

Extract an orthonormal basis from the columns of a matrix.

Batched factorizations

Process multiple matrices stored in an array with leading batch dimensions.

Interview questions

QWhat does numpy.linalg.qr compute, and what does the default return?

It decomposes a matrix into Q and R, where Q has orthonormal columns and R is upper triangular. The default reduced mode uses K = min(M, N), producing Q with shape (..., M, K) and R with shape (..., K, N).

Follow-up: How would you obtain only the triangular factor? Use mode='r', which returns R with shape (..., K, N), avoiding construction of Q.

QHow do reduced and complete modes differ?

The complete mode returns Q with shape (..., M, M) and R with shape (..., M, N). For a real input Q is orthogonal; for a complex input it is unitary.

Follow-up: What does complete mode imply for a square real matrix? Q is a square orthogonal matrix, so its columns and rows are orthonormal.

QWhat does mode='raw' return?

The raw mode returns h and tau instead of Q and R. h stores Householder reflectors and tau stores their scaling factors; their dtypes are np.double or np.cdouble.

Follow-up: Why might raw mode be useful? It exposes the factorization's reflector representation when downstream code needs the underlying Householder data rather than explicit Q.

numpy.linalg.lstsq
numpy.linalg.svd
numpy.linalg.cholesky
numpy.linalg.solve

Source: numpy.linalg.qr. Checked 2026-10-15.